Mathematics vs Physics (1960-1969)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulas, are the beginning of every physical theory." (Leopold Infeld,"The Evolution of Physics", 1961)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future." (Paul A M Dirac , "The Evolution of the Physicist’s Picture of Nature ", Scientific American, 1963)

"We wish to see [...] the typical attitude of the scientist who uses mathematics to understand the world around us [...] In the solution of a problem [...] there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude. (George Pólya, "Mathematical Methods in Science", 1963)

"In its efforts to learn as much as possible about nature, modem physics has found that certain things can never be ‘known’ with certainty. Much of our knowledge must always remain uncertain. The most we can know is in terms of probabilities." (Richard P Feynman,"The Feynman Lectures on Physics", 1964)   

"It bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what a tiny piece of space-time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple, like the checker board with all its apparent complexities." (Richard P Feynman, "The Character of Physical Law", 1965)

"Pedantry and sectarianism aside, the aim of theoretical physics is to construct mathematical models such as to enable us, from the use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. Any logically sound theory satisfying this condition is a good theory, whether or not it be derived from 'ultimate' or 'fundamental' truth. It is as ridiculous to deride continuum physics because it is not obtained from nuclear physics as it would be to reproach it with lack of foundation in the Bible." (Clifford Truesdell & Walter Noll, "The Non-Linear Field Theories of Mechanics", 1965)

"When the problems in physics become difficult we may often look to the mathematician who may already have studied such things and have prepared a line of reasoning for us to follow. On the other hand they may not have, in which case we have to invent our own line of reasoning, which we then pass back to the mathematician." (Richard Feynman,"The Character of Physical Law", 1965)

"Mathematicians, on the other hand, often regard all of physics as a kind of divine revelation or trickery, where mathematical morals are irrelevant, so that if they enter this red-light district at all, it is only to get what they want as cheaply as possible before returning to the respectability of problems purely mathematical in the older sense: analysis, probability, differential geometry, etc." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"It is impossible, and it has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form." (Jacob Klein, "Greek Mathematical Thought and the Origin of Algebra", 1968)

Mathematics vs Physics (1950-1959)

"Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience." (Albert Einstein, "Out of My Later Years", 1950)

"Automata have begun to invade certain parts of mathematics too, particularly but not exclusively mathematical physics or applied mathematics. The natural systems (e.g., central nervous system) are of enormous complexity and it is clearly necessary first to subdivide what they represent into several parts that to a certain extent are independent, elementary units. The problem then consists of understanding how these elements are organized as a whole. It is the latter problem which is likely to attract those who have the background and tastes of the mathematician or a logician. With this attitude, he will be inclined to forget the origins and then, after the process of axiomatization is complete, concentrate on the mathematical aspects." (John Von Neumann, "The General and Logical Theory of Automata", 1951)

"In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus 'really' not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time." (Hermann Weyl, "Space, Time, Matter", 1952)

"The principal mathematical element in the culture, embodying the living and growing mass of modern mathematics, will be chiefly possessed by the professional mathematicians. True, certain professions, such as engineering, physics, and chemistry, which employ a great deal of mathematics, carry a sizable amount of the mathematical tradition, and in some of these, as in the case of physics and engineering research, some individuals contribute to the growth of the mathematical element in the culture. But, in the main, the mathematical element of our culture is dependent for its existence and growth on the class of those individuals known as ‘mathematicians’." (Raymond L Wilder, "Introduction to the Foundations of Mathematics", 1952)

"There are at least four fundamental purposes that the study of mathematics should attain. First, it should serve as a functional tool in solving our individual everyday problems. [...] In the second place, mathematics serves as a handmaiden for the explanation of the quantitative situations in other subjects, such as economics, physics, navigation, finance, biology, and even the arts. [...] In the third place, mathematics, when properly conceived, becomes a model for thinking, for developing scientific structure, for drawing conclusions, and for solving problems. [...] In the fourth place, mathematics is the best describer of the universe about us." (Howard F Fehr, "Reorientation in Mathematics Education", Teachers Record 54, 1953)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies--so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects. (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)

"To the author the main charm of probability theory lies in the enormous variability of its applications. Few mathematical disciplines have contributed to as wide a spectrum of subjects, a spectrum ranging from number theory to physics, and even fewer have penetrated so decisively the whole of our scientific thinking." (Mark Kac, "Lectures in Applied Mathematics" Vol. 1, 1959)

Mathematics vs Physics (1925-1949)

"Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover." (Bertrand Russell, "An Outline of Philosophy", 1927)

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised [...]" (Arthur S Eddington, "Science and the Unseen World", 1929)

"The present tendency of physics is toward describing the universe in terms of mathematical relations between unimaginable entities." (John W N Sullivan, "The Bases of Modern Science", 1929)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"The chain of cause and effect could be quantitatively verified only if the whole universe were considered as a single system - but then physics has vanished, and only a mathematical scheme remains. The partition of the world into observing and observed system prevents a sharp formulation of the law of cause and effect." (Werner K Heisenberg, "The Physical Principles of the Quantum Theory", 1930)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"[...] the mathematical physicist [...] obtains much prestige from the physicists because they are impressed with the amount of mathematics he knows, and much prestige from the mathematicians, because they are impressed with the amount of physics he knows." (William F G Swann, "The Architecture of the Universe", 1934)

"Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation." (Paul A M Dirac, "The Relation Between Mathematics And Physics", Proceedings of the Royal Society of Edinburgh", 1938-1939)

"Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. [...] Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society of Edinburgh, 1938-1939)

"There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology." (Paul A M Dirac, [Lecture delivered on presentation of the James Scott prize] 1939)

"At the present time it is of course quite customary for physicists to trespass on chemical ground, for mathematicians to do excellent work in physics, and for physicists to develop new mathematical procedures […] Trespassing is one of the most successful techniques in science." (Wolfgang Köhler, "Dynamics in Psychology", 1940)

"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts." (Ernst Mach, "The Science of Mechanics" 5th Ed, 1942)

"Physicists who are trying to understand nature may work in many different fields and by many different methods; one may dig, one may sow, one may reap. But the final harvest will always be a sheaf of mathematical formulae. These will never describe nature itself, hut only our observations on nature. Our studies can never put us into contact with reality; we can never penetrate beyond the impressions that reality implants in our minds." (James H Jeans,"Physics and Philosophy" 3rd Ed., 1943)

"Good physics is made a priori. Theory precedes fact. Experience is useless because before any experience we are already in possession of the knowledge we are seeking for. Fundamental laws of motion (and of rest), laws that determine the spatio-temporal behavior of material bodies, are laws of a mathematical nature. Of the same nature as those which govern relations and laws of figures and numbers. We find and discover them not in Nature, but in ourselves, in our mind, in our memory, as Plato long ago has taught us." (Alexander Koyre, "Galileo and the Scientific Revolution of the Seventeenth Century", The Philosophical Review Vol. 52 (3), 1943)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

Mathematics vs Physics (1900-1924)

"The science of physics does not only give us [mathematicians] an opportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré, "La valeur de la science" ["The Value of Science"], 1905)

"If we turn to the problems to which the calculus owes its origin, we find that not merely, not even primarily, geometry, but every other branch of mathematical physics - astronomy, mechanics, hydrodynamics, elasticity, gravitation, and later electricity and magnetism - in its fundamental concepts and basal laws contributed to its development and that the new science became the direct product of these influences. [...] The calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word." (William Osgood,"The Calculus in Colleges in Colleges and Technical Schools", Bulletin of the American Mathematical Society, 1907)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought." (Ernest W Hobson, [address] 1910)

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"It is characteristic of modern physics to represent all processes in terms of mathematical equations. But the close connection between the two sciences must not blur their essential difference." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

"The physical object cannot be determined by axioms and definitions. It is a thing of the real world, not an object of the logical world of mathematics. Offhand it looks as if the method of representing physical events by mathematical equations is the same as that of mathematics. Physics has developed the method of defining one magnitude in terms of others by relating them to more and more general magnitudes and by ultimately arriving at 'axioms', that is, the fundamental equations of physics. Yet what is obtained in this fashion is just a system of mathematical relations. What is lacking in such system is a statement regarding the significance of physics, the assertion that the system of equations is true for reality." (Hans Reichenbach, "The Theory of Relativity and A Priori Knowledge", 1920)

"[...] the future of thought and therefore of history lies in the hands of physicists, and therefore the future historian must seek his education in the world of mathematical physics." (Henry Adams, "The Degradation of the Democratic Dogma", 1920)

"It is undeniable that mathematicians, with a self-denying ordinance about coefficients, can thus attain remarkable criteria, and are able to anticipate definite results; but we need not seek to engraft their modes of expression on the real world of physics." (Oliver J Lodge, "Geometrization of Physics, and Its Supposed Bias on the, Michelson-Morley Experiment", Nature Vol. 106 (2677), 1921)

🅼M Gustav Mittag-Leffler - Collected Quotes

"How enticing science now seems to me! How lovely knowledge must be! In its richly objective sphere, one could forget the self in order to dissolve into ecstasy. Perhaps I don’t have enough brains to be a scientist, to understand thoroughly what is studied. Yet my deepest wish is to be able to do so, and one should always try." (M Gustav Mittag-Leffler)

"Mathematics, when untouched by applications to other sciences, can easily be diverted onto some of the obscure paths of formalism and philosophizing." (M Gustav Mittag-Leffler)

"Number is the beginning and the end of thought. With thought, number is born. Without number, thought goes nowhere." (M Gustav Mittag-Leffler)

"Our will and testament has come about out of a vigorous conviction that a nation that does not highly esteem mathematical thought can never be capable of achieving the highest cultural goals and thereby enjoy the international respect which, in the long term, is an effective means of maintaining our position in the world, as well as asserting our right to live our own lives." (M Gustav Mittag-Leffler)

"The best and highest purpose in my life has been the cultivation of the science which is the most abstract and most cosmopolitan of all, which less than any other is nourished by roots in its own homeland." (M Gustav Mittag-Leffler)

"The mathematician's best work is art […] a high and perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch each other." (M Gustav Mittag-Leffler)

"The reason for the usual but highly superficial view of mathematics that perceives this field of study as demanding more special talents than the study of any other science is the fact that all other school subjects can be learned superficially, as merely dead lessons learned by rote, but that this is nearly impossible with mathematics." (M Gustav Mittag-Leffler)

"Yes, it’s unity that we first and foremost need to strive for in our world, as far as that is possible. That is why a scientist also has to avoid becoming involved in politics. The task of science has to be to unite." (M Gustav Mittag-Leffler)

On Impossibility (Unsourced)

"Anytime new insight replaces an old assumption, or a fossilized perception is the spring. New understandings sprout, new tolerances appear, and new curiosity draws you to previously dark places. Just as the sun shines earlier and longer in the spring, changes that seemed impossible appear to be possible with each new insight into your own health." (Gary Zukav)

"I did not understand how such a quantity could be real, when imaginary or impossible numbers were used to express it." (Gottfried W Leibniz)

"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible." (Carl Friedrich Gauss)

"In the world of human thought generally, and in physical science particularly, the most important and fruitful concepts are those to which it is impossible to attach a well-defined meaning." (Hendrik A Kramers)

"Indeed, when in the course of a mathematical investigation we encounter a problem or conjecture a theorem, our minds will not rest until the problem is exhaustively solved and the theorem rigorously proved; or else, until we have found the reasons which made success impossible and, hence, failure unavoidable. Thus, the proofs of the impossibility of certain solutions plays a predominant role in modern mathematics; the search for an answer to such questions has often led to the discovery of newer and more fruitful fields of endeavour." (David Hilbert)

"It often happens that understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solution. (Freeman J Dyson)

"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem." (Howard W Eves)

"It is impossible to transcend the laws of nature. You can only determine that your understanding of nature has changed." (Nick Powers)

"It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset." (Sir Arthur S Eddington)

"Mathematical language, precise and adequate, nay, absolutely convertible with mathematical thought, can afford us no example of those fallacies which so easily arise from the ambiguities of ordinary language; its study cannot, therefore, it is evident, supply us with any means of obviating those illusions from which it is itself exempt. The contrast of mathematics and philosophy, in this respect, is an interesting object of speculation; tut, as imitation is impossible, one of no practical result." (Sir William R Hamilton)

"String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory." (Edward Witten)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." (Carl Friedrich Gauss)

"The universe has no circumference, for if it had a center and a circumference there would be some and some thing beyond the world, suppositions which are wholly lacking in truth. Since, therefore, it is impossible that the universe should be enclosed within a corporeal center and corporeal boundary, it is not within our power to understand the universe, whose center and circumference are God. And though the universe." (Nicholas of Cusa)

On Impossibility (-1599)

"A likely impossibility is always preferable to an unconvincing possibility. The story should never be made up of improbable incidents; there should be nothing of the sort in it." (Aristotle, "Poetics", cca. 335 BC)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one." (Aristotle, "Physics", cca. 350 BC)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle, "Physics", cca. 350 BC)

[...] to repeat the same throw ten thousand times with the dice would be impossible, whereas to make it once or twice is comparatively easy. (Aristotle, "On the Heavens", cca. 350 BC)

"Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible." (Pappus of Alexandria, cca. 4th century BC)

"Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible." (Archimedes, "On Spirals", cca. 225 BC)

"When, therefore, as will be clear to those who read, the passage as a connected whole is literally impossible, whereas the outstanding part of it is not impossible but even true, the reader must endeavor to grasp the entire meaning, connecting by an intellectual process the account of what is literally impossible with the parts that are not impossible but historically true, these being interpreted allegorically in common with the part which, so far as the letter goes, did not happen at all. For our contention with regard to the whole of divine scripture is that it all has a spiritual meaning, but not all a bodily meaning; for the bodily meaning is often proved to be an impossibility." (Origen Adamantius, "On First Principles", cca. 220-230)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"So in all human affairs one notices, if one examines them closely, that it is impossible to remove one inconvenience without another emerging." (Niccolò Machiavelli, "Discourses on Livy", 1531)

"A second type of the false position makes use of roots of negative numbers. I will give an example: If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion. This, however, is closest to the quantity which is truly imaginary since operations may not be performed with it as with a pure negative number, nor as in other numbers. [...] This subtlety results from arithmetic of which this final point is, as I have said, as subtle as it is useless." (Girolamo Cardano, "Ars Magna", 1545)

"Given that annihilation of nature in its entirety is impossible, and that death and dissolution are not appropriate to the whole mass of this entire globe or star, from time to time, according to an established order, it is renewed, altered, changed, and transformed in all its parts." (Giordano Bruno, "The Ash Wednesday Supper", 1584)

On Impossibility (1600-1699)

"Another argument of hope may be drawn from this–that some of the inventions already known are such as before they were discovered it could hardly have entered any man's head to think of; they would have been simply set aside as impossible. For in conjecturing what may be men set before them the example of what has been, and divine of the new with an imagination preoccupied and colored by the old; which way of forming opinions is very fallacious, for streams that are drawn from the springheads of nature do not always run in the old channels." (Sir Francis Bacon, "Novum Organum", 1620)

"But by far the greatest obstacle to the progress of science and to the undertaking of new tasks and provinces therein is found in this - that men despair and think things impossible." (Sir Francis Bacon, "Novum Organum", 1620)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, "The Assayer", 1623)

"I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said 'it appears to us', because I am not against thinking that not even for such an effect does Nature make use of straight line motion." (Galileo Galilei, [Letter to Francesco Ingoli] 1624)

"Someone could also ask what these impossible solutions are. I would answer that they are good for three things: for the certainty of the general rule, for being sure that there are no other solutions, and for its utility." (Albert Girard, "L'Invention nouvelle de l'Algébre", 1629)

"Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632)

"But it is just that the Roots of Equation should be impossible, lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)

"We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible." (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"From a given determined cause an effect follows of necessity, and on the other hand, if no determined cause is granted, it is impossible that an effect should follow." (Baruch Spinoza, "Ethics", 1677)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, "Treatise of Algebra", 1685)

"It is impossible for a Die, with such determin’d force and direction, not to fall on such a determin’d side, only I don’t know the force and direction which makes it fall on such a determin’d side, and therefore I call that Chance, which is nothing but want of Art [...]" (John Arbuthnot, "Of the Laws of Chance", 1692)

On Impossibility (1700-1749)

"From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised." (Gottfried W Leibniz, "Specimen novum analyses pro Scientia infinity circa summas et quadraturas", 1700)

"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)

"It is your opinion, the ideas we perceive by our senses are not real things, but images, or copies of them. Our knowledge therefore is no farther real, than as our ideas are the true representations of those originals. But as these supposed originals are in themselves unknown, it is impossible to know how far our ideas resemble them; or whether they resemble them at all. We cannot therefore be sure we have any real knowledge." (George Berkeley, "Three Dialogues", 1713)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz, "Monadology", 1714)

"By the very nature of poetry it is impossible for everyone to be at the same time a sublime poet and a sublime metaphysician, for metaphysics abstracts the mind from the senses, and the poetic faculty must submerge the whole mind in the senses. Metaphysics soars up to universals, and the poetic faculty must plunge deep into particulars." (Giambattista Vico, "The New Science", 1725)

"But it is just that the Roots of Equations should be often impossible (complex), lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton,"Universal Mathematic" 2nd Ed., 1728)

 "[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination." (Leonhard Euler, 1732)

"A problem was posed to me about an island in the city of Königsberg, surrounded by a river spanned by seven bridges, and I was asked whether someone could traverse the separate bridges in a connected walk in such a way that each bridge is crossed only once. I was informed that hitherto no-one had demonstrated the possibility of doing this, or shown that it is impossible. This question is so banal, but seemed to me worthy of attention in that not geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether or not such a round trip is possible […]" (Leonard Euler, [letter to Giovanni Marinoni] 1736)

"But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal." (David Hume,"Treatise of Human Nature", 1738)

"For it ought to be considered that both –b and –c , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b) and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination." (Thomas Simpson,"A Treatise of Algebra", 1745)

"Man is so complicated a machine that it is impossible to get a clear idea of the machine beforehand, and hence impossible to define it. For this reason, all the investigations have been vain, which the greatest philosophers have made à priori, that is to say, in so far as they use, as it were, the wings of the spirit. Thus it is only à posteriori or by trying to disentangle the soul from the organs of the body, so to speak, that one can reach the highest probability concerning man's own nature, even though one can not discover with certainty what his nature is." (Julien Offray de La Mettrie, "Man a Machine", 1747)

On Impossibility (1750-1799)

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"As a general rule - never substitute the symbol for the thing signified, unless it is impossible to show the thing itself; for the child's attention is so taken up with the symbol that he will forget what it signifies." (Jean Jacques Rousseau, "Emile, or On Education", 1762)

"One of the most intimate of all associations in the human mind is that of cause and effect. They suggest one another with the utmost readiness upon all occasions; so that it is almost impossible to contemplate the one, without having some idea of, or forming some conjecture about the other." (Joseph Priestley, "The History and Present State of Electricity", 1767)

"Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination." (Leonhard Euler, "Vollständige Anleitung zur Algebra", 1768-69)

"All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible." (Euler, Algebra, 1770)

"But ignorance of the different causes involved in the production of events, as well as their complexity, taken together with the imperfection of analysis, prevents our reaching the same certainty about the vast majority of phenomena. Thus there are things that are uncertain for us, things more or less probable, and we seek to compensate for the impossibility of knowing them by determining their different degrees of likelihood. So it was that we owe to the weakness of the human mind one of the most delicate and ingenious of mathematical theories, the science of chance or probability." (Pierre-Simon Laplace, "Recherches, 1º, sur l'Intégration des Équations Différentielles aux Différences Finies, et sur leur Usage dans la Théorie des Hasards", 1773)

"That metaphysics has hitherto remained in so vacillating a state of uncertainty and contradiction, is only to be attributed to the fact, that this great problem, and perhaps even the difference between analytical and synthetical judgements, did not sooner suggest itself to philosophers. Upon the solution of this problem, or upon sufficient proof of the impossibility of synthetical knowledge a priori, depends the existence or downfall of metaphysics. (Immanuel Kant, "Critique of Pure Reason" , 1781)

"Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed […] Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"We must therefore establish a form of decision-making in which voters need only ever pronounce on simple propositions, expressing their opinions only with a yes or a no. […] Clearly, if anyone’s vote was self-contradictory (intransitive), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible." (Nicolas de Condorcet, "On the form of decisions made by plurality vote", 1788)

"It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept a word is needed; to portray a phenomenon a concept is needed. All three mirror one and the same reality." (Antoine-Laurent Lavoisier, "Traite Elementaire de Chimie", 1789)

"The impossibility of separating the nomenclature of a science from the science itself is owing to this, that every branch of physical science must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact." (Antoine-Laurent de Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary. Hence they talk of two roots to every equation of the second order, and the learner is to try which will succeed in a given equation: they talk of solving an equation which requires two impossible roots to make it solvable: they can find out some impossible numbers, which, being multiplied together, produce unity. This is all jargon, at which common sense recoils; but, from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust, and hate the labour of a serious thought." (William Frend, "The Principles of Algebra", 1796)

"Certain authors who seem to have perceived the weakness of this method assume virtually as an axiom that an equation has indeed roots, if not possible ones, then impossible roots. What they want to be understood under possible and impossible quantities, does not seem to be set forth sufficiently clearly at all. If possible quantities are to denote the same as real quantities, impossible ones the same as imaginaries: then that axiom can on no account be admitted but needs a proof necessarily." (Carl F Gauss, "New proof of the theorem that every algebraic rational integral function in one variable can be resolved into real factors of the first or the second degree", 1799)